A Matrix that defines a Graph or a Network. $A$

$A_{ij} = 1 \text{ if edge (j,i) exists}$. $A_{ij} = 0 \text{ if edge (j,i) doesn't exist}$

$A^T=A$ if undirected.

$A$ describes same network if we permute columns and rows in the same way.

Weighted adjacency matrix (or weight matrix) $W$ assigns a weight to edges. Usually weight is a real number: $w: E \rightarrow \mathbb{R}$

"Topology" represented by $A$.

"Geometry represented by $W$.

Related: Graph laplacian